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Theorem resf2nd 16321
Description: Value of the functor restriction operator on morphisms. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
resf1st.f (𝜑𝐹𝑉)
resf1st.h (𝜑𝐻𝑊)
resf1st.s (𝜑𝐻 Fn (𝑆 × 𝑆))
resf2nd.x (𝜑𝑋𝑆)
resf2nd.y (𝜑𝑌𝑆)
Assertion
Ref Expression
resf2nd (𝜑 → (𝑋(2nd ‘(𝐹f 𝐻))𝑌) = ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)))

Proof of Theorem resf2nd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-ov 6527 . 2 (𝑋(2nd ‘(𝐹f 𝐻))𝑌) = ((2nd ‘(𝐹f 𝐻))‘⟨𝑋, 𝑌⟩)
2 resf1st.f . . . . . 6 (𝜑𝐹𝑉)
3 resf1st.h . . . . . 6 (𝜑𝐻𝑊)
42, 3resfval 16318 . . . . 5 (𝜑 → (𝐹f 𝐻) = ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩)
54fveq2d 6089 . . . 4 (𝜑 → (2nd ‘(𝐹f 𝐻)) = (2nd ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩))
6 fvex 6095 . . . . . 6 (1st𝐹) ∈ V
76resex 5347 . . . . 5 ((1st𝐹) ↾ dom dom 𝐻) ∈ V
8 dmexg 6963 . . . . . 6 (𝐻𝑊 → dom 𝐻 ∈ V)
9 mptexg 6364 . . . . . 6 (dom 𝐻 ∈ V → (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) ∈ V)
103, 8, 93syl 18 . . . . 5 (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) ∈ V)
11 op2ndg 7046 . . . . 5 ((((1st𝐹) ↾ dom dom 𝐻) ∈ V ∧ (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) ∈ V) → (2nd ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))))
127, 10, 11sylancr 693 . . . 4 (𝜑 → (2nd ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))))
135, 12eqtrd 2640 . . 3 (𝜑 → (2nd ‘(𝐹f 𝐻)) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))))
14 simpr 475 . . . . . 6 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
1514fveq2d 6089 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd𝐹)‘𝑧) = ((2nd𝐹)‘⟨𝑋, 𝑌⟩))
16 df-ov 6527 . . . . 5 (𝑋(2nd𝐹)𝑌) = ((2nd𝐹)‘⟨𝑋, 𝑌⟩)
1715, 16syl6eqr 2658 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd𝐹)‘𝑧) = (𝑋(2nd𝐹)𝑌))
1814fveq2d 6089 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
19 df-ov 6527 . . . . 5 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
2018, 19syl6eqr 2658 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝑋𝐻𝑌))
2117, 20reseq12d 5302 . . 3 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)) = ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
22 resf2nd.x . . . . 5 (𝜑𝑋𝑆)
23 resf2nd.y . . . . 5 (𝜑𝑌𝑆)
24 opelxpi 5059 . . . . 5 ((𝑋𝑆𝑌𝑆) → ⟨𝑋, 𝑌⟩ ∈ (𝑆 × 𝑆))
2522, 23, 24syl2anc 690 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑆 × 𝑆))
26 resf1st.s . . . . 5 (𝜑𝐻 Fn (𝑆 × 𝑆))
27 fndm 5887 . . . . 5 (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆))
2826, 27syl 17 . . . 4 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
2925, 28eleqtrrd 2687 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom 𝐻)
30 ovex 6552 . . . . 5 (𝑋(2nd𝐹)𝑌) ∈ V
3130resex 5347 . . . 4 ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)) ∈ V
3231a1i 11 . . 3 (𝜑 → ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)) ∈ V)
3313, 21, 29, 32fvmptd 6179 . 2 (𝜑 → ((2nd ‘(𝐹f 𝐻))‘⟨𝑋, 𝑌⟩) = ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
341, 33syl5eq 2652 1 (𝜑 → (𝑋(2nd ‘(𝐹f 𝐻))𝑌) = ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3169  cop 4127  cmpt 4634   × cxp 5023  dom cdm 5025  cres 5027   Fn wfn 5782  cfv 5787  (class class class)co 6524  1st c1st 7031  2nd c2nd 7032  f cresf 16283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-2nd 7034  df-resf 16287
This theorem is referenced by:  funcres  16322
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